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2 Triangle intersection Many, many ways to do this Most robust (and one of the fastest) is to do it based on determinants For vectors a,b,c define Det(a,b,c)=±6 volume(tet(a,b,c)), the signed volume of the tetrahedron spanned by edges a,b,c from a common point Sign flips when tetrahedron reflected, or alternatively from right-hand-rule on a bc Triangle intersection boils down to 2 sign checks: segment crosses plane 3 sign checks: line goes through triangle
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3 Triangle Mesh (more) Object normal Normalize cross-product of two sides of the triangle Distance from single triangle Find barycentric coordinates -- solve a least- squares problem Need to clip to sides of triangle Compute distance from that point Note: also gives direction to closest point Distance (and direction) from mesh Compute for all possible triangles, take minimum Trick is to find small list of possible triangles with acceleration structure
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4 Implicit Surface Simple function, metaballs, or interpolated from 3d grid (“level set”) Recall - for metaballs need acceleration Particle inside: f(x)<0 Trajectory cross: Just like ray-tracing - use secant method Object normal: f/| f | Distance from surface: If f() is signed distance, then trivial Otherwise, painful, but f() might be good enough for application
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5 Back to particle collisions So now we can represent other geometry, how do we do a repulsion velocity field? v(x)=f(distance(x)) * n(x) n(x) is the outward direction (=normal on surface) f is some decreasing function that drops towards zero far away Exponential f(d)=e -k*d Or linear drop, truncated to zero: f(d)=max(0,m-k*d) Or more complicated Outward direction is plus/minus direction to closest point Aside: useful for more than just collisions - e.g. fire particles streaming out of an object
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6 Force-based repulsions Can do exactly the same trick for force- based motion Add repulsion field to F(x) Simple, often works, but there are sometimes problems What are you trying to model? Robustness - high velocity impacts can penetrate arbitrarily far High velocity impacts may go straight through thin objects How much of a rebound do you want?
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7 Damped repulsions Think of repulsion force as a generalized spring Add spring damping: D is some parameter you set n(x) is the outward direction again
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8 Aside: springs and damping How do you come up with reasonable values for spring constants and damping constants? And how do you pick good step sizes for differential equation solver (Forward Euler etc.) Look at 1D simplified model Ma=F=-Kx-Dv M is the mass, K is like a spring stiffness, D is the damping parameter Solve it analytically
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9 Critical Damping Three cases: Underdamped (D 2 -4MK<0) Oscillation with frequency Characteristic time: Exponentially decays at rate Characteristic time: Overdamped (D 2 -4MK>0) No continued oscillation Exponentially decays at rates Characteristic times: Critically damped (D 2 -4MK=0) No continued oscillation Fastest decay possible at rate Characteristic time:
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10 Numerical time steps Should be proportional to minimum characteristic time Implicit methods like Backwards Euler actually let you take larger steps with stability, but wipe out all hope of accuracy for things with small characteristic time For nonlinear multi-dimensional forces, what are K and D? Estimate them by figuring out what is the fastest |F| can change if you modify x or v respectively This is all very approximate, so don’t get hung up on getting the “right” answer Will ultimately need a fudge factor anyhow (from experiments)
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11 True Collisions Turn attention from repulsions for a while Model collision as a discrete event - a bounce Input: incoming velocity, object normal Output: outgoing velocity Need some idea of how “elastic” the collision Fully elastic - reflection Fully inelastic - sticks (or slides) Let’s ignore friction for now Let’s also ignore how to incorporate it into algorithm for moving particles for now
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12 Newtonian Collisions Say object is stationary, normal at point of impact is n Incoming particle velocity is v Split v into normal and tangential components: Newtonian model for outgoing velocity Unchanged tangential component v T New normal component is The “coefficient of restitution” is , ranging from 0 (inelastic) to 1 (perfectly elastic) The final outgoing velocity is
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13 Relative velocity in collisions What if particle hits a moving object? Now process collision in terms of relative velocity v rel =v particle -v object Take normal and tangential components of relative velocity Reflect normal part appropriately to get new v rel Then new v particle =v object +(new v rel )
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